Impulse Response Function Estimation by EM: Estimating DV and DVR with No Model Assumption

11. Impulse response function estimation by EM -Estimating DV and DVR with no model at all. The only assumption made is that the response of the tissues T and R (‘target’ and ‘reference’) to the plasma input P is linear and shift invariant (the response to two injections is the sum of their individual responses, the response to twice the input is twice the response, it doesn’t matter what time the injection is) Thus the response function is the input function convolved with the impulse response function, the response of the system to a perfect delta function input.

12. Reference tissue methods, whether graphical or compartmental, appear to assume that the reference tissue time course is the input function to the target tissue. What’s really going on is that the reference tissue time course is used to infer what the unmeasured plasma input function must have been. In the reference tissue model, the target tissue is modeled as the convolution of the reference tissue with the target-reference (as opposed to the target-plasma) impulse response function. Strange but true: the target-plasma IRF is the reference-plasma IRF convolved with the target-reference IRF

13. The integral over all time of the tissue-plasma IRF is the distribution volume of that tissue: It follows therefore that the integral over all time of the target-reference IRF is the ratio of the distribution volumes (DVR), the outcome of the tissue input Logan method.

14. Now digitize the problem: The same routine works for any one of the three situations already described, plasma-target, plasma-reference, or reference-target. Ti is the time course being fitted, Ci is the input function time course, either plasma or reference tissue, and Ii is the relevant impulse response function.

15. If we know Ti and we know Ci, we can estimate what Ii must have been using the principle of Expectation-Maximization: • Make initial guess Ij (r labels the iteration) • Compute the target time course Ti that this Ij would yield • Compute the ratio of the measured time course and model time course at each time point i

16. Update each Ij according to the composite opinion of the ratios Qi at each time point i, with each individual opinion weighted by Ci-j+1 The sum in the coefficient denominator normalizes the Ci-j+1 Repeat

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